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82       XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

(c)    Let Jt* be the monoid consisting of the adjoints U* of the operators

then Jt* is weakly compact (Problem 9(d)). Show that R(X) and F(^*) are ortho-
gonal supplementary vector subspaces of E. (Use (b) to reduce to the case where Jt is
commutative.) Deduce that F(^) is stable under Jt.

(d)    Let (AC()i*n be a finite sequence of vectors in E. For each /, let xt  yi + zt ,
where yt  R(^) and zt e F(*O. Use (c) to show that there exists U e Jt such that
U - zt  0 (1 < i ^ n), and then prove that there exists Ue^f such that VU - yi = yt
for 1 ^ / <; TZ. (Consider the space E" and the image ^' of Jf in -$?(E"; E") consisting
of the operators (?i, . . . , f)i-(!r- fi, . . ., 7*- /) where Te ^, and use the fact that
RMO is a vector subspace.) Deduce that the set P(#i, . . . , xn) of operators WzJt
such that W - xt ~ W  yt = yt for 1 ^i^n is not empty, and then that the inter-
section of all these sets (where (xi) varies over all finite sequences of vectors in E)
is  nonempty (Section 12.3, Problem 6(e)).   Finally, deduce that the  projection
P : E > R(~O corresponding to the direct sum decomposition E = R(M)  F(^)
belongs to ^ and hence to ^, and use Problem 10 to conclude that R(^) and F(*^)
are orthogonal supplementary vector subspaces (" Jacobs' theorem ").

(e)    Show that for each C/e^T there exists Ve^f such that VU - y*=y for all
y e R(^) (same method as in (d)). In other words, the image ^ of ^ under the map-
ping U\-+ {7|R(~O is a subgroup of the unitary group U(R(~O). Show that the orbits
of this subgroup are strongly compact in R(^) (cf. Problem 3(c)) and deduce that ^ is
strongly compact in .S?(R(JO; R(^0),

(f)   Let (en)neZ be an orthonormal basis of E, and let Jt be the weakly compact
submonoid of *$ generated by the unitary operator U defined by U ' - en  en+1 for
all n e Z. Show that F(^) = E and R(^)  {0}.

12.   In the notation of Problem 9, let J( be any submonoid of #, containing IE- For
each x e E, let T(J? - x) be the closed convex hull of the orbit of x under ^.

(a)    Show that T(^f - x) contains a ^-invariant point (cf. Problem 3 (a)). Deduce
that, if $(<^) denotes the set of ^-invariant vectors and &(~O the set of x e E such
that 0 e T(J(  x\ then every vector in E can be written as the sum of a vector
belonging to O(^) and a vector belonging to l(<Jf).

(b)    Show that <(~^) and H(^*) are orthogonal supplementary vector subspaces
of E. Use Problem 10 to deduce that <D(~O and Q.(^f) are orthogonal supplementary
vector subspaces of E. Deduce that T(J? - x) contains only one -^-invariant point.

(c)   Deduce from (b) that, for each contraction U e tf and each vector x e E, the
limit

1  n-l

lim -  Uk-x
n-+oo n fc*= i

exists and is invariant under U (" von Neumann's ergodic theorem ").

(a) Let E be a vector space over R or C, and let V be a vector space of linear forms
on E, endowed with the weak topology. Let A be a weakly closed convex set in E,
such that 0 g A. Show that there exists a weakly closed hyperplane H in V passing
through 0, such that H is the set of all /e V for which /(JCG) = 0 for some JCG 7* 0 in E,
and such that g(xG) |> 0 for all g e A. (Use Problem 1 to reduce to the case where V
is finite-dimensional, by observing that there exists a neighborhood of 0 containing a
closed subspace W of finite codimension in V, and such that the projection of A on a
supplement L of W in V does not meet a neighborhood of 0 in L. Then apply the result
of Problem 3(a).)#.